In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between al...In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between algicidal bacteria and algae. Firstly, mathematical work mainly provided some threshold conditions to ensure the occurrence of transcritical bifurcation and saddle-node bifurcation, which could provide certain theoretical support for selecting key ecological environmental factors and numerical simulations. Secondly, the numerical simulation work dynamically displayed the evolution process of the bifurcation dynamic behavior of the model (2.1) and the growth coexistence mode of algae and algicidal bacteria. Finally, it was worth summarizing that intrinsic growth rate and combined capture effort of algae population had a strong influence on the dynamic behavior of the model (2.1). Furthermore, it must also be noted that transcritical bifurcation and saddle-node bifurcation were the inherent driving forces behind the formation of steady-state growth coexistence mode between algicidal bacteria and algae. In summary, it was hoped that the results of this study would contribute to accelerating the study of the interaction mechanism between algicidal bacteria and algae.展开更多
Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt...Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt.The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and bifurcations.The equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation methods.The results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the branches.In addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.展开更多
The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf...The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.展开更多
The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlin...The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.展开更多
In this paper n-dimensional flows (described by continuous-time system) with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalis...In this paper n-dimensional flows (described by continuous-time system) with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalism. Based on frequency domain approach, we prove some criterions for the saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and give an example to illustrate the efficiency of the result obtained.展开更多
In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynam...In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate <em>r</em> and searching efficiency <em>a</em>. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for <em>b≠a</em> where <em>a,b</em> are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.展开更多
This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibil...This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is obtained.We show that the system exhibits transcritical bifurcation.To show the existence of Bogdanov-Takens bifurcation,we have derived the normal form.We have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient evanescences.To show the existence of saddle-node bifurcation,we used Sotomayor's theorem.Furthermore,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure.An optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.展开更多
The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the pred...The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the predation process,which benefits the predators by developing fear upon their prey.This study investigates the dynamical behavior of a modified LV-type predator-prey system with Michaelis-Menten-type harvesting of predators where predators adopt cooperation strategy during hunting.The ecologically feasible steady states of the system and their asymptotic stabilities are explored.The local codimension one bifurcations,viz.transcritical,saddle-node and Hopf bifurcations,that emerge in the system are investigated.Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation.A backward Hopfbifurcation is detected when the harvesting effort is increased,which destabilizes the system by generating periodic solutions.The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient.Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations.Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased.We have also explored codimension two bifurcations viz.the generalized Hopf and the Bogdanov-Takens bifurcation exhibit by the system.To visualize the dynamical behavior of the system,numerical simulations are conducted using an ecologically plausible parameter set.The existence of the bionomic equilibrium of the model is analyzed.Moreover,an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.展开更多
This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system ca...This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system can be approximated by two simplified systems S and F, which correspond respectively to slow and fast subsystems. The singularities, including Hopf bifurcation (HB), saddle-node bifurcation (SNB) and singularity induced bifurcation (SIB), are characterized. We show that SNB occurs at P Tc = 3.4382, SIB at P T0 = 2.8653 and HB at P Th = 2.802 for the singular perturbation system. It means that the power system will collapse near SIB which precedes SNB and that the power system will oscillate near HB which precedes SIB. In other words, the power system will lose its stability by means of oscillation near the HB which precedes SIB and SNB as P T is increasing to a critical value. The boundary of the stability region of the system can be described approximately by a combination of boundaries of the stability regions of the fast subsystem and slow subsystem.展开更多
In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equ...In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equilibria of the model.In our model,the HTCs are stimulated by the identification of the presence of tumor antigens.Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells,which leads to the elimination of the bistable behavior of the tumor-immune system,i.e.the equilibrium corresponding to the high steady state of tumor cells is destabilized.Choosing immune intensity c as bifurcation parameter,there exists saddle-node bifurcation.Besides,there exists a critical value C*,at which a Hopf bifurcation occurs.The stability and direction of Hopf bifurcation are discussed.展开更多
文摘In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between algicidal bacteria and algae. Firstly, mathematical work mainly provided some threshold conditions to ensure the occurrence of transcritical bifurcation and saddle-node bifurcation, which could provide certain theoretical support for selecting key ecological environmental factors and numerical simulations. Secondly, the numerical simulation work dynamically displayed the evolution process of the bifurcation dynamic behavior of the model (2.1) and the growth coexistence mode of algae and algicidal bacteria. Finally, it was worth summarizing that intrinsic growth rate and combined capture effort of algae population had a strong influence on the dynamic behavior of the model (2.1). Furthermore, it must also be noted that transcritical bifurcation and saddle-node bifurcation were the inherent driving forces behind the formation of steady-state growth coexistence mode between algicidal bacteria and algae. In summary, it was hoped that the results of this study would contribute to accelerating the study of the interaction mechanism between algicidal bacteria and algae.
基金financial support from K.N.Toosi University of Technology,Tehran,Iran。
文摘Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt.The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and bifurcations.The equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation methods.The results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the branches.In addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.
文摘The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.
基金the National Natural Science Foundation of China (60574011)Department of Science and Technology of Liaoning Province (2001401041).
文摘The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.
基金This work was supported by the National Natural Science Foundation of China (No. 10371136).
文摘In this paper n-dimensional flows (described by continuous-time system) with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalism. Based on frequency domain approach, we prove some criterions for the saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and give an example to illustrate the efficiency of the result obtained.
文摘In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate <em>r</em> and searching efficiency <em>a</em>. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for <em>b≠a</em> where <em>a,b</em> are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.
基金The authors also thankfully acknowledge financial support from Council of Scientific and Industrial Research,India through a research fellowship(File No.09/013(0841)/2018-EMR-I)Jyoti Maurya and DST-Science and Engineering Research Board,MATRICS Expert committee(File No.MTR/2021/000819)A.K.Misra to carry out this research work.
文摘This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is obtained.We show that the system exhibits transcritical bifurcation.To show the existence of Bogdanov-Takens bifurcation,we have derived the normal form.We have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient evanescences.To show the existence of saddle-node bifurcation,we used Sotomayor's theorem.Furthermore,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure.An optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.
基金jointly supported by the National Natural Science Foundation of China(62173139)the Science and Technology Innovation Program of Hunan Province(2021RC4030).
文摘The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the predation process,which benefits the predators by developing fear upon their prey.This study investigates the dynamical behavior of a modified LV-type predator-prey system with Michaelis-Menten-type harvesting of predators where predators adopt cooperation strategy during hunting.The ecologically feasible steady states of the system and their asymptotic stabilities are explored.The local codimension one bifurcations,viz.transcritical,saddle-node and Hopf bifurcations,that emerge in the system are investigated.Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation.A backward Hopfbifurcation is detected when the harvesting effort is increased,which destabilizes the system by generating periodic solutions.The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient.Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations.Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased.We have also explored codimension two bifurcations viz.the generalized Hopf and the Bogdanov-Takens bifurcation exhibit by the system.To visualize the dynamical behavior of the system,numerical simulations are conducted using an ecologically plausible parameter set.The existence of the bionomic equilibrium of the model is analyzed.Moreover,an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.
基金Supported by the National Natural Science Fundation of China (No.50377018)a research grant from Research Office of the Hong Kong Polytechnic University(G.63.37.T494)
文摘This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system can be approximated by two simplified systems S and F, which correspond respectively to slow and fast subsystems. The singularities, including Hopf bifurcation (HB), saddle-node bifurcation (SNB) and singularity induced bifurcation (SIB), are characterized. We show that SNB occurs at P Tc = 3.4382, SIB at P T0 = 2.8653 and HB at P Th = 2.802 for the singular perturbation system. It means that the power system will collapse near SIB which precedes SNB and that the power system will oscillate near HB which precedes SIB. In other words, the power system will lose its stability by means of oscillation near the HB which precedes SIB and SNB as P T is increasing to a critical value. The boundary of the stability region of the system can be described approximately by a combination of boundaries of the stability regions of the fast subsystem and slow subsystem.
文摘In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equilibria of the model.In our model,the HTCs are stimulated by the identification of the presence of tumor antigens.Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells,which leads to the elimination of the bistable behavior of the tumor-immune system,i.e.the equilibrium corresponding to the high steady state of tumor cells is destabilized.Choosing immune intensity c as bifurcation parameter,there exists saddle-node bifurcation.Besides,there exists a critical value C*,at which a Hopf bifurcation occurs.The stability and direction of Hopf bifurcation are discussed.